Some time ago, as I began reading a history of mathematics, I was surprised and amused to find that there are African cultures alive and well today, in the modern age, that simply have no concept of mathematics. I don't mean that they don't recognize the formula for the area of a circle. I mean that they have no concepts for numbers greater than two. After two they literally begin using "two-and-one", "two-and-two", "two-and-two-and-one", and so forth. If you bought one sheep for two bags of rice, the shepherd would smile and be happy, but if you tried to buy two sheep for four bags of rice, he'd look confused and wonder if he was being cheated. The concepts just aren't there.

Eventually it was proved that if one geometry was consistent, then all the other ones must be as well. And while no culture in history ever began exploring geometry from a non-Euclidean point of view, it's intriguing to think of what would happen if one did. (The Pythagorean Theorem would look vastly different, for a start.)

Mathematics is not "real". It's not something that was ever discovered, it was invented and reinvented throughout history as a way of explaining the physical world using abstract symbols. Logic, you could argue, is immutable and culturally objective. Same goes for physics. But these are not mathematics, and should never be confused with it.

Euclid knew what he was doing. He didn't call the fifth postulate an "axiom", he called it a "postulate". He suspected it should be a logical consequence of the axioms, but was unable to prove it. This makes sense -- we know that Euclid was wrong, and that it is indeed independent (what Euclid would call "an axiom").

In any case, he realised that he needed it, and that he could not prove it, so he it stated outright. In particular, this means that everything in The Elements implicitly depends on it; Euclid did not hide this fact. And if you're performing e.g. spherical geometry on the half sphere, you can immediately see that Euclid's results do not necessarily follow, since the fifth postulate does not hold on a sphere.

And, in fact, Eratosthenes (who came from roughly the same culture, although later) did do spherical geometry! And guess what -- he didn't make use of results in Euclidian geometry that aren't true on a sphere! This did not interfere with his work (he measured the diameter of the Earth based on empirical measurements conducted in Egypt). He was able to do all of this because of the fundamental nature of mathematical law: it is true in some domain, and that domain is precisely defined. (The same is not true of physics, where the nature of the approximation can give rise to problems).

So you see Euclid's results are absolute truths: IF you accept the axioms and postulates, THEN the sum of the angles in a triangle is 2right angles, there are precisely 5 regular polyhedra, the laws of similarity (homotheticity) hold, etc. And if you do not accept the axioms and postulates (e.g. because you're trying to navigate a ship, and you know the fifth postulate does not hold), then none of these results follows (although they might still be true, and even provable by some other argument).

Let's go back to the result about the sum of angles in a triangle. It does not hold without the fifth postulate. Then again, it does not hold for quadrangles, pentagons, hexagons or any other polygon. This does not mean that its truth is relative or depends on a cultural context. It merely means that it is in the nature of theorems that the result does not necessarily follow if the assumptions do not hold. And that is not surprising.

In fact, we even see that culture, which supposedly "forced" Euclid to consider only plane geometry with straightedge and compass, was no match for mathematics: Euclid knew he had to state his assumptions precisely; his Elements is based on taking this set of assumptions as far as was possible in his age.